THE ASTROPHYSICAL JOURNAL, 485 : 680È688, 1997 August 20
( 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.
MAGNETOHYDRODYNAMIC TURBULENCE REVISITED
P. GOLDREICH1 AND S. SRIDHAR2
Received 1995 September 5 ; accepted 1997 March 25
ABSTRACT
In 1965, Kraichnan proposed that MHD turbulence occurs as a result of collisions between oppositely
directed Alfven wave packets. Recent work has generated some controversy over the nature of nonlinear
couplings between colliding Alfven waves. We Ðnd that the resolution to much of the confusion lies in
the existence of a new type of turbulence, intermediate turbulence, in which the cascade of energy in the
inertial range exhibits properties intermediate between those of weak and strong turbulent cascades.
Some properties of intermediate MHD turbulence are the following : (1) in common with weak turbulent
cascades, wave packets belonging to the inertial range are long-lived ; (2) however, components of the
strain tensor are so large that, similar to the situation in strong turbulence, perturbation theory is not
applicable ; (3) the breakdown of perturbation theory results from the divergence of neighboring Ðeld
lines due to wave packets whose perturbations in velocity and magnetic Ðelds are localized, but whose
perturbations in displacement are not ; (4) three-wave interactions dominate individual collisions between
wave packets, but interactions of all orders n º 3 make comparable contributions to the intermediate
turbulent energy cascade ; (5) successive collisions are correlated since wave packets are distorted as they
follow diverging Ðeld lines ; (6) in common with the weak MHD cascade, there is no parallel cascade of
energy, and the cascade to small perpendicular scales strengthens as it reaches higher wavenumbers ; (7)
for an appropriate weak excitation, there is a natural progression from a weak, through an intermediate,
to a strong cascade.
Subject headings : ISM : general È MHD È turbulence
1.
INTRODUCTION
the density and transport properties of the Ñuid are constant, the equations of (incompressible) MHD read
There appears to be some consensus regarding the notion
that turbulence in the ionized interstellar medium is of magnetohydrodynamic origin. However, even three decades
after Iroshnikov (1963) and Kraichnan (1965) Ðrst, independently, presented their ideas on MHD turbulence (in an
incompressible, highly conducting Ñuid), there is much
debate on what this theory really is. Recent controversy has
centered on the existence, or otherwise, of certain nonlinear
interactions among Alfven waves. Sridhar & Goldreich
(1994, hereafter SG) have argued that there are no resonant
three-wave interactions in weak MHD turbulence. They
also asserted that the Iroshnikov-Kraichnan (IK) theory is
incorrect and constructed a theory of weak MHD turbulence based on resonant four-wave interactions ; in this
theory, nonlinear interactions strengthen on small spatial
scales, resulting ultimately in a strong cascade proposed by
Goldreich & Sridhar (1995). However, Montgomery &
Matthaeus (1995) have maintained that resonant threewave interactions are nonempty, holding these responsible
for the anisotropic cascade seen in the numerical simulations of Shebalin, Matthaeus, & Montgomery (1983). Moreover, Ng & Bhattacharjee (1996, hereafter NB) have
recently presented calculations that show that smallamplitude wave packets do interact via three waves, so long
as the k \ 0 (zü is the direction of the local, mean magnetic
z
Ðeld) Fourier
components of velocity and magnetic Ðeld
perturbations are nonzero. The present investigation is an
analysis and resolution of this controversy.
We consider magnetic turbulence in an incompressible
Ñuid, although we simply call it MHD turbulence. When
L b \ $ Â (¿ Â b) ] i+2b ,
t
L ¿ \ [(¿ Æ $)¿ ] (b Æ $)b [ $p ] c+2¿ ,
t
$ Æ ¿\$ Æ b\0 ,
(1a)
(1b)
(1c)
where ¿ is the velocity, b \ B/(4no)1@2 is the magnetic Ðeld
in velocity units, and p is the ratio of total pressure to the
density. The dissipation provided by i and c is important
only on small spatial scales, so we ignore the dissipative
terms. The homogeneous state, ¿ \ 0, B \ B zü , is a
0 Alfven
stable, static solution of equations 0(1a)È(1c).0 Shear
waves and pseudo-Alfven waves are the two kinds of linear
perturbations about this equilibrium, the latter being the
incompressible limit of the slow magnetosonic wave. Both
kinds of waves have the same dispersion relation, namely,
u \ v o k o, where v \ B /(4no)1@2 is called the Alfven
z perturbed
A velocity
0
speed.A The
and magnetic Ðelds are
related by d¿ \ ^db, where the upper/lower signs correspond to waves traveling antiparallel/parallel to B (with
0 with
k \ 0 and k [ 0, respectively). Equations (1a)È(1c),
z
z
i \ c \ 0, possess the remarkable property of allowing for
nonlinear generalizations of the linear Alfven waves.
Mutual cancellation of nonlinear terms permits the following wide class of exact solutions : if d¿(x) \ [db(x) at some
instant of time, t \ 0, it can be checked that d¿(x, y, z
[ v t) \ [db(x, y, z [ v t) for all time, irrespective of the
A
A Parker 1979). This nonlinear
functional
form of d¿(x) (see
solution describes a wave packet of arbitrary form traveling
nondispersively in the direction of B . Similarly, we can also
construct another class of nonlinear 0solutions, with d¿ \ db,
that travels nondispersively in a direction opposite to B .
0
Both types of nonlinear solutions are stable, and the
dynamics is simple so long as there is no spatial overlap
(““ collisions ÏÏ) between oppositely moving wave packets.
1 Division of Geological and Planetary Sciences, California Institute of
Technology, Pasadena, CA 91125 ; pmg=nicholas.caltech.edu.
2 Inter-University Centre for Astronomy and Astrophysics, Post Bag 4,
Ganeshkhind, Pune 411 007, India ; sridhar=iucaa.ernet.in.
680
MHD TURBULENCE REVISITED
Let us consider a situation that is less restrictive than
perturbations about a static equilibrium (a uniform magnetic Ðeld). We imagine that turbulent motions on a large
scale are set up in the Ñuid by stirring it in a random but
statistically steady fashion. These create large-scale, disordered velocity and magnetic Ðelds. Kinetic and magnetic
energies will cascade to smaller spatial scales.3 The central
problem of MHD turbulence is to determine the statistical
steady state amplitudes of the Ñuctuations in ¿ and b, on
intermediate spatial scalesÈthe so-called inertial-range
power spectrum. Mathematically speaking, this cascade
through spatial scales should emerge from the e†ect of the
nonlinear terms in equations (1a)È(1c). Physically, it is
common to speak of interactions between ““ eddies ÏÏ (see
Frisch 1995). Do interactions between eddies of dissimilar
sizes make signiÐcant contributions to the form of the
cascade ? For hydrodynamic turbulence, the answer seems
to be that no, the dominant interactions are between eddies
of similar spatial scales. The reason for such a locality in
interactions between eddies is that the sweeping due to the
velocity Ðeld of a large-sized eddy (on a smaller eddy) may
be transformed away by a local Galilean transformation.
Kraichnan noted that, for MHD turbulence, such a transformation has no e†ect at all on the magnetic Ðeld of the
large eddy ; the magnetic Ðeld of a large eddy acts upon
smaller eddies in much the same manner as the mean magnetic Ðeld does on Alfven wave packets. Hence MHD turbulence in an incompressible Ñuid should reduce to turbulence of
interacting Alfve n wave packets. We recall from the previous
paragraph that Ðnite-amplitude wave packets that travel in
the positive/negative z-directions do so without change in
form, so long as oppositely directed packets do not overlap.
Kraichnan realized that the cascade of energy in MHD turbulence occurs as a result of collisions between oppositely
directed Alfve n wave packets.
To derive the results of the IK theory, consider a statistically steady, isotropic excitation of amplitude v > v ,
l
A
on outer scale l, of the static equilibrium mentioned earlier.
Alfven waves have dv D db, so that v is the amplitude of
l
excitation of both velocity and magnetic
Ðelds. The
resulting Ñuctuations may, at any time, be decomposed into
Alfven wave packets with scales j [ l traveling in the positive and negative z-directions. Iroshnikov and Kraichnan
assumed that the energy transfer is local and isotropic in
k-space. Collisions between oppositely directed wave
packets occur over times of order u~1 D (v k)~1, and these
k collision,
A
create small distortions. During one
each wave
packet su†ers a fractional perturbation
dv
v
dv
j D j (v kv )~1 D j > 1 .
(2)
dt A j
v
v
A
j
During successive collisions, these perturbations add with
random phases. The number of collisions for the fractional
perturbations to build up to order unity is
A B AB
v 2
v 2
(3)
N D j D A .
j
dv
v
j
j
The energy cascade rate is v D v2/t , where the cascade time
j 2).
j The three-dimensional
is given by t D N /(v k) D v /(kv
j
j
A
A
j
energy spectrum, E(k), is related to the velocity Ñuctuation
3 On scales small enough for the i+2b and the c+2¿ terms to be important, kinetic and magnetic energies will dissipate into heat.
681
by v2 D k3E(k). Using KolmogorovÏs hypothesis of the scale
j
independence of v, we obtain the following scalings for the
inertial range of the IK theory :
AB
j 1@4
v
jD
,
l
v
l
With these scalings
E(k) D
v2
l
.
l1@2k7@2
A BAB
v 2 l 1@2
,
N D A
j
v
j
l
so the cascade weakens as j decreases.
2.
(4)
(5)
INTERMEDIATE TURBULENCE
2.1. A New Cascade Based on T hree-W ave Interactions
SG argued that the IK theory, although seemingly plausible, was basically incorrect. Here we discuss the reason for
the failure of this theory and propose a new cascade based
on three-wave interactions. Let us begin by listing three key
features of the IK theory :
1. Wave packets of size j live for N wave periods ; large
j
values of N correspond to weak interactions
between wave
j
packets of size j. From equations (3) and (4), it may be
veriÐed that N P 1/j1@2, so that the cascade weakens as it
progresses intojthe inertial range.
2. From equation (2), the fractional perturbation su†ered
by a wave packet during one collision is D(v /v ) > 1 ; in
A
the IK theory, interactions between wave jpackets
are
described by the lowest order nonlinear terms.
3. Isotropy is assumed in the derivation of the IK theory.
The derivation of this theory is essentially heuristic, roughly
as given above in equations (2)È(4). We note this so that it is
clear to the reader that there is not a more rigorous version
of the ““ theory ÏÏ that we happened not to mention.
Together, features 1 and 2 imply that the ““ elementary
interactions ÏÏ between Alfven waves must satisfy the threewave resonance conditions
k ]k \k , u ]u \u ,
(6)
1
2
3
1
2
3
where u \ v o k o. Shebalin et al. (1983) noted that the only
k solutions
A z
nontrivial
of equations (6) require that one of
either k or k must be zero. This implies that waves with
1,zk not
2,z present initially cannot be created during
values of
z
collisions between oppositely directed wave packets. As
they point out, there is no parallel (i.e., along k ) cascade of
z
energy. Thus the turbulence must be anisotropic,
and
energy should cascade to large k .
In what follows, we derive anM anisotropic version of IK
theory. As before, we imagine that the system is stirred in a
statistically steady and isotropic fashion such that v > v
l
A
on outer scale l. The absence of a parallel cascade implies
that wave packets belonging to the inertial range have
parallel scales l and perpendicular scales j > l. We estimate
the spectrum of the anisotropic cascade by modifying the
arguments of equations (2)È(4) so as to keep track of parallel
and perpendicular scales. In one collision between two such
oppositely directed wave packets, the fractional perturbation is given by
dv
lv
dv
j D j (v k v )~1 D j > 1 .
A
z
j
dt
jv
v
A
j
(7)
682
GOLDREICH & SRIDHAR
Adding up the perturbations due to successive collisions
with random phases, the number of collisions over which
the perturbations grow to order unity is
A B A B
v 2
jv 2
A .
N D j D
j
dv
lv
j
j
Making use of the steady state relation
(8)
v2 v2
j D l Dv
(9)
t
t
l
j
(““ KolmogorovÏs hypothesis ÏÏ), we obtain scaling relations
for the velocity Ñuctuations, as well as the threedimensional energy spectrum of the anisotropic, three-wave
cascade :
AB
j 1@2
v2
v
jD
, E(k , k ) D l .
z M
l
k3
v
M
l
It follows that
for o z o ? l. The distance along z over which o * o increases by
a factor of order unity from its initial value of j at z \ 0 is
A B
jv 2
l
A lD
L D
;
(14)
*
lv
s2
j
L is a function of j. The signiÐcance of L in intermediate
*
*
turbulence follows because turbulence involves the transfer
of energy across scales. If s > 1, single interactions between
wave packets result in small perturbations. Cascading of
energy requires of order
L
1
N D D *?1
j s2
l
(15)
such interactions.
2.2.2. Nonlinear Interactions in Intermediate T urbulence
(10)
AB
v 2j
,
(11)
N D A
j
v
l
l
so nonlinearity increases along the inertial range of the new
cascade.
The anisotropic cascade di†ers from the original IK
cascade in (1) being anisotropic, (2) having a di†erent spectrum, and (3) strengthening at high wavenumber. This Ðnal
di†erence has a profound consequence. It implies that the
anisotropic cascade has a limited inertial range, thereby
diminishing its applicability in astronomical contexts,
where the excitation at the outer scale is likely to be quite
strong.
It turns out that this anisotropic cascade is an example of
a new type of turbulence, which we call intermediate turbulence, because it has properties intermediate between those
of weak and strong turbulence. In particular, intermediate
turbulence does not submit to perturbation theory.
2.2. T he Failure of Perturbation T heory in Intermediate
T urbulence
2.2.1. Field-L ine Geometry
To lowest order in perturbation theory, wave packets
move along Ðeld lines. Thus the breakdown of perturbation
theory may be understood physically by studying the
geometry of the divergence of a bundle of Ðeld lines. Assume
that the mean Ðeld lies along the z-axis. Consider wave
packets localized in velocity and magnetic Ðeld perturbations, but not in displacement, having longitudinal scale l
and transverse scale j with j/l \ 1. Let us require that the
velocity Ñuctuation, v , is small enough,
j
lv
(12)
s4 j >1 ,
jv
A
so that N ? 1 (cf. eq. [8]). The rms di†erential inclination
j Ðeld across scale j is h D v /v . It is correlated
of the local
j
j A to z.
over distances l and j parallel and orthogonal
Let us focus on a pair of neighboring Ðeld lines separated
by j at z \ 0. The expectation value of the separation
between these Ðeld lines, *, varies such that
*2 D j2 ] h2 l o z o
j
Vol. 485
(13)
We have deduced the form of the spectrum of intermediate MHD turbulence from scaling arguments based on
three-wave couplings. Moreover, these interactions are
weak in the sense that N ? 1. This might suggest that
j
three-wave interactions dominate
those of higher order and
that a rigorous derivation of the steady state cascade might
result from truncation at this order. Unfortunately, this is
incorrect ; interactions of all orders have similar strengths.
Consider the distortion su†ered during a single collision
between oppositely directly wave packets of similar strength
with perpendicular and parallel dimensions j [ l and l. We
assume that lv > jv . Contributions from interactions
j mayAbe written as
involving n waves
A B
lv n~2
dnv
j
jD
.
(16)
jv
v
A
j
Clearly, three-wave interactions dominate those of higher
order for individual collisions.
Next we consider the cumulative distortion due to
n-wave interactions as a wave packet travels a distance l >
z > L . We can picture the distortion as arising from the
* of the packet as it follows the di†erential wandershearing
ing of neighboring Ðeld lines.4 The net displacement of individual Ðeld lines over distance z deÐnes a vector Ðeld whose
shear tensor transforms the packetÏs shape.5 For l > z > L
*
this transformation is close to the identity, so it may be
expanded in a Taylor series. The expansion parameter is
(z/L )1@2, the dimensionless measure of fractional spreading
over* distance z of a bundle of Ðeld lines whose crosssectional radius at z \ 0 is j. Terms of order n [ 2 in v
correspond to n-wave interactions.6 These terms have thej
form
A B AB
A B
lv n~2 z (n~2)@2
z (n~2)@2
dnv
j
jD
D
.
(17)
jv
l
L
v
A
*
j
A notable feature of equation (17) is that contributions from
higher order (n º 4) interactions carry extra factors of
(z/l)1@2. By ““ extra factors,ÏÏ we mean those beyond the single
factor (z/l)1@2 expected to arise from the addition of a
sequence of independent interactions between pairs of wave
4 A uniform displacement of a wave packet does not contribute to the
energy cascade.
5 This transformation is subject to the constraints of Ñuid incompressibility and magnetic Ñux freezing.
6 Since wave packets follow Ðeld lines only to lowest order, nonkinematic terms appear at orders n º 4. Those of n \ 4 are given in ° 3.3.2.
No. 2, 1997
MHD TURBULENCE REVISITED
packets. These factors reveal an interdependence among
collisions associated with the nonlocalized Ðeld-line displacements of the wave packets.7
Intermediate turbulence is nonperturbative because distortions of all orders become large as z ] L . This is as
*
expected, because the cascade time across scale j is t D
j
L /v . It implies that n-wave interactions of all orders n º 3
* A
make comparable contributions to the energy cascade.
A Ñuid element whose Lagrangian coordinate is a has
Eulerian location x, at time t, given by
(18)
where n(a, t) is the displacement Ðeld. Velocity and magnetic
Ðeld perturbations at the Eulerian location are deÐned in
terms of the displacement vector by
Ln
Ln
(a, t) , b(x, t) \ v
(a, t) .
(19)
A La
Lt
A
SG employed a formulation of the MHD action due to
Newcomb (1962) and developed a Lagrangian perturbation
theory for weak MHD turbulence. The Lagrangian is a
functional of the strain tensor Ðeld, in other words, the
gradient of the displacement vector Ðeld. Expansion of the
Lagrangian density in powers of the strain tensor yields
terms of second order, n \ 2, and then fourth and higher
orders, n º 4. The absence of third-order terms signiÐes that
wave packets follow Ðeld lines to lowest nonlinear order.
The absence of Lagrangian perturbations based on threewave interactions implies that Eulerian perturbations due
to three-wave interactions are purely kinematic. Kinematic
contributions to Eulerian perturbations are also present at
each higher order, n º 4. But these are augmented by
dynamic perturbations arising from order n º 4 terms in
the expansion of the Lagrangian.
As before, we consider small-amplitude, (lv /jv ) > 1,
l
wave packets localized in ¿ and b but not in n.j Roughly
speaking, convergence of the perturbative expansion
requires the components of the strain tensor to be smaller
than unity.8 An ensemble of independent wave packets generates an energy spectrum that is Ñat for k l [ 1. Since u \
z
v o k o, the power spectrum of the displacement
vector Ðeld
A
z
varies as k~2 for k l > 1. The same behavior characterizes
the power zspectra zof some of the components of the strain
tensor. It implies that these components diverge. The divergence is the mathematical expression of the spreading of
Ðeld-line bundles described in ° 2.2.1. In this light, the failure
of perturbation theory is seen to be generic, and not just a
consequence of an unfortunate choice of perturbation variable.
Next we investigate the e†ect of cutting o† the energy
spectrum below k L D 1, where L Z l. The most strongly
z
divergent components
of mij have power spectra given by
¿(x, t) \
o m8ij(k) o2 D
AB
v 2 l
j
.
v
k2
A
z
Multiplying equation (20) by k2 and integrating from k D
M
z
L~1 to k D l~1, we obtain the average value of o mij o2 due to
z
power in this wavenumber interval,
o mij o2 D j~2
P
A B
l~1
lv 2 L
j
dk o m8ij(k) o2 D
.
z
jv
l
L~1
A
(21)
Thus
o mij o2 D L /L
2.2.3. L agrangian Perturbation T heory
x \ a ] n(a, t) ,
683
(20)
7 These extra factors of (z/l)1@2 are absent in the corresponding formula
for wave packets that are localized in displacement.
8 From this point on it is best to proceed in Fourier space, since the
breakdown of perturbation theory is closely related to the behavior of the
energy spectrum (i.e., velocity, or magnetic Ðeld power spectra) at small k .
z
,
(22)
*
where L is deÐned in equation (14). Once again we see the
*
crucial role played by L ; the perturbative expansion con*
verges if the energy spectrum is cut o† below k L D 1.
z *
The absence of third-order terms in the expansion
of the
Lagrangian signiÐes the absence of resonant three-wave
interactions in perturbative MHD turbulence. Weak MHD
turbulence based on four-wave interactions is discussed in
° 3.2.
3.
ON WEAK AND INTERMEDIATE TURBULENCE
3.1. T ypes of T urbulence
We have discussed, at some length, the intermediate
cascade. It is time to state in a precise manner the properties
that characterize the three kinds of turbulence. Let us begin
with a deÐnition. Weak turbulence is characterized by the
following properties :9
1. Nonlinear interactions among an ensemble of waves
that are weak, in the sense that the fractional change in
wave amplitude during each wave period is small.
2. The existence of a convergent perturbative expansion
for the nonlinear interactions. Typically the small parameter is the fractional change in wave amplitude during a
wave period.
When these two conditions are satisÐed, a formal theory of
resonant wave interactions may be derived. For the theory
to be nonempty, either the three-wave or four-wave resonance relations must possess nontrivial solutions. Power
spectra of cascades arise as stationary solutions of the
kinetic equation describing modal energy transfer.
SpeciÐcally, for MHD turbulence, we Ðnd the following :
a) The turbulent cascade based on four-wave interactions, derived in SG, is the unique weak cascade that
satisÐes both conditions 1 and 2 above. Moreover, this
cascade is realizable ; it could in principle be set up experimentally. While three-wave interactions do not vanish, they
are nonresonant and do not transfer energy among di†erent
waves.
b) The critically balanced cascade described in Goldreich
& Sridhar (1995) is an example of strong turbulence. It violates both conditions 1 and 2. The interaction time is of the
order of the wave period. Interactions of all orders have
comparable strengths ; there is no valid perturbative expansion.
c) There is a third type of MHD turbulence that satisÐes
condition 1 but not condition 2. It is an example of what we
have called intermediate turbulence. This turbulence exhibits
weak interactions, but strains in the Ñuid are so strong that
perturbation theory diverges. The three-wave interactions
are included, but not dominant ; it turns out that inter9 For a standard discussion of these points, see the introduction to
Zakharov, LÏvov, & Falkovich (1992).
684
GOLDREICH & SRIDHAR
Vol. 485
actions of all orders contribute equally weakly. The inertialrange spectrum of intermediate MHD turbulence is given,
for the Ðrst time, in equation (10) of this paper.
the weak turbulence of SG is realizable (see ° 4 below), this
feature makes it less applicable than intermediate turbulence.
3.2. W eak Alfve nic T urbulence
Having discussed intermediate turbulence in some detail,
we provide a brief outline of the theory of weak MHD
turbulence that SG constructed using resonant four-wave
interactions. By studying the resonant terms of the fourthorder Lagrangian, SG derived a formal kinetic equation for
the evolution of energies (more precisely, ““ wave action ÏÏ) in
di†erent modes and proved that a cascade of energy indeed
emerges as a stationary solution. The elementary interactions involve scattering of two waves that respect the
following conservation laws :
3.3. A Controversy and Its Resolution
NB have proved, analytically, that three-wave interactions between small-amplitude wave packets are nonzero
if the k \ 0 Fourier components are nonzero. On the other
z
hand, SG have argued, using a Lagrangian perturbation
theory, that three-wave interactions are absent in weak
MHD turbulence. In ° 2, we claimed that the interactions
found by NB lead to intermediate, rather than weak, turbulence. Here we bolster that claim by an explicit evaluation
of three-wave and four-wave interactions in Lagrangian
coordinates.
The simplest derivation employs the Lagrangian displacement vector Ðeld as the basic variable (cf. eqs. [18] and
[19]). Incompressibility implies that the transformation
between Lagrangian and Eulerian coordinates have unit
Jacobian. Thus
k ]k \k ]k , u ]u \u ]u .
(23)
1
2
3
4
1
2
3
4
Using u \ v o k o and the z-component of the equation
z SG proved that k \ k [ 0 and
involvingk theA kÏs,
1,zof equation
3,z
k \ k \ 0. Of course, the symmetry
(23)
2,z
4,z
allows us to Ñip the signs of all the k Ïs, or permute indices 3
and 4 ; the important point is thatz the scattering process
described by equation (23) leaves the k -components unalz of k not present
tered. This implies that waves with values
z
in the external stirring cannot be created by resonant
fourwave interactions.
In addition to developing a formal theory, SG also provided a heuristic derivation of the weak four-wave cascade
for shear Alfven waves. Here we note the main properties of
the weak cascade of shear Alfven waves :
1. As discussed above, there is no transfer of energy to
small spatial scales in the z-direction ; the energy cascade in
k-space occurs only along k (i.e., in directions perpendicuM
lar to the mean magnetic Ðeld).
2. The three-dimensional energy spectrum, E, is deÐned
by
; v2 \
j
P
d3k
E(k , k )
,
z M 8n3
(24)
where the sum is over wave packets of various scales. Weak
turbulence relies on a convergent perturbation theory. As
discussed in ° 2.2.3, this requires that the spectrum, E, be cut
o† for o k L o \ 1. Moreover, weak four-wave interactions
z * k , which implies that E may have a quite
do not change
z
arbitrary dependence
on o k o. This simply depends on the
nature of the excitation andz is not of much interest here. If
j D k~1 is a perpendicular length scale belonging to the
M range, the scalings derived by SG for the weak,
inertial
four-wave cascade are
AB
j 2@3
v2
v
l
jD
, E(k , k ) D
.
(25)
z
M
l
l1@3k10@3
v
M
l
3. The number of collisions needed for the packet to lose
memory of its initial state is
J \ 1 ] $ Æ n [ 1 $n : $n ] 1 ($ Æ n)2 ] 1 mijmjkmki
2
2
3
[ 1 ($ Æ n)mijmji ] 1 ($ Æ n)3 ,
(27)
2
6
where $ refers to derivatives with respect to Lagrangian
coordinates and
$n : $n 4 mijmji .
The displacement vector n is split into transverse and
longitudinal components g and f, respectively, such that
n\g]f ,
(29)
$ Æ g\0 .
(30)
with
The components of g are the two independent variables ; f is
obtained from equation (27). Thus
(31)
$ Æ f \ 1 $g : $g ,
1
2 2 1
(32)
$ Æ f \ $g : $g ] $g : $f [ 1 gijgjkgki .
3
1
2
1
2 3 1 1 1
The Ðrst term on the right-hand side of equation (32) is
included for completeness, since g \ 0.10
Following the development in2 ° 3 of SG, we write the
Lagrangian as
P AK K
K KB
Ln 2
Ln 2
[ v2
.
(33)
A La
Lt
A
However, we present our calculations in real space, rather
than in Fourier space ; the results are identical, but the realspace version turns out to be useful later. Solving equation
(31) yields
L\
A B A BAB
jv 4
v 4 j 4@3
A D A
.
(26)
N D
j
lv
v
l
j
l
Note that, in common with the spectrum of intermediate
turbulence (given in eq. [10]), N decreases as the cascade
j
proceeds to smaller j.
4. In their treatment of weak turbulence, SG unwittingly
made the assumption that E was cut o† at small o k o. While
z
(28)
o
2
d3a
f \ [$
2
P
d3a@ $g : $g
1
1.
8n o a [ a@ o
(34)
We now write
L\L ]L
2
4
10 See eq. (42) and the footnote that follows it.
(35)
No. 2, 1997
MHD TURBULENCE REVISITED
(the third-order terms vanish), with
o
L \
2 2
o
L \
4 2
P AK K
P AK K
d3a
d3a
Variation of the action
S4
P
K KB
K KB
Lg 2
Lg 2
1 [ v2
1
,
A La
Lt
A
Lf 2
Lf 2
2 [ v2
2
.
A La
Lt
A
dt[L ]
P
d3aP($ Æ g)]
(36)
(37)
(38)
with respect to g leads to the (Euler-Lagrange) equation of
motion.11
3.3.1. T hree-W ave Interactions in L agrangian Coordinates
In this subsection, we recover the results of the threewave interactions calculated by NB. Moreover, we demonstrate that they are purely kinematic in Lagrangian
coordinates.
To lowest order, the contribution of L may be ignored.
4 of S (eq. [38])
Using equation (36) for L in the variation
2 linear equation :12
results in the following simple,
A
B
L2
L2
[ v2
g \0 ,
(39)
A La2 1
Lt2
A
whose general solution is a superposition of wave packets
traveling in the positive and negative z-directions :
g \ g`(a , a [ v t) ] g~(a , a ] v t) .
(40)
1
1 M A
A
1 M A
A
NB calculated, in Eulerian coordinates, the lowest order
perturbation due to a collision between oppositely directed
wave packets. It is a trivial matter to obtain this quantity by
using Lagrangian perturbation theory. To do so, we transform the right-hand side of the expression for ¿, given in
equation (19), into Eulerian coordinates. To Ðrst order,
Lg
¿ (x, t) \ 1 (x, t)
1
Lt
(41)
is the velocity Ðeld of unperturbed wave packets, and g is
the corresponding displacement Ðeld. To second order, 1
Lf
¿ \ [(g Æ $ )¿ ] 2 ,
2
1
x 1
Lt
(42)
where the subscript x refers to Eulerian coordinates.13
When we substitute equation (40) in the right-hand side of
equation (42), we obtain three di†erent types of terms ; those
that contain two powers of g` or g~ have nothing to do
1 collisions.
1
with perturbations induced by
Only the mixed
terms describe distortions su†ered by a wave packet during
a collision with an oppositely directed wave packet. If
*gB \ <[gB(x , ]O) [ gB(x , [O)]
(43)
1
1 M
1 M
is the net displacement of Ðeld lines due to the (^) wave
packets, then the asymptotic distortion due to a collision is
*¿B \ [(*gY Æ $ )¿B [ $
2
1
x 1
x
P
d3x@ $ (*gY) : $ ¿B
x 1
x 1 .
4n
o x [ x@ o
11 P is a Lagrange multiplier needed to ensure that $ Æ g \ 0.
12 We assign Ðrst order to g of unperturbed wave trains.
13 The absence of Lg /Lt is due to the vanishing of L .
2
3
(44)
685
The above expression is equivalent to equations (15) and
(16) of NB. The wave packet distortions expressed by equation (44) are kinematic, as described in ° 2.2.3. Displacements resulting from any sequence of collisions are
obtained by summing individual values.14 Note that both
terms on the right-hand side of equation (44) depend on
*gB, the net displacement of Ðeld lines, which is related to
1
the Fourier amplitude of gB with k \ 0.
1
z
3.3.2. Four-W ave Interactions in L agrangian Coordinates
Our principal aim in this subsection is to demonstrate
that four-wave interactions obey the distance scaling proposed in equation (17). Our secondary goals are to obtain
the spectrum of weak Alfvenic turbulence from a
conÐguration-space calculation and to recover the
frequency-changing terms discovered by NB, in full
MHD.15
We have shown that the three-wave interactions of NB
arise from kinematic perturbations in Eulerian coordinates.
Dynamic perturbations require the interaction of at least
four waves and are associated with perturbations in
Lagrangian coordinates. We already possess the machinery
necessary to derive these. When L is included in the action
4 respect to g leads to
of equation (38), the variation with
1
Euler-Lagrange equations, which may be thought
of as
adding third-order terms to the right-hand side of equation
(39) :
A
B
A
B
L2
L2
L2
L2
$P3
[ v2
g \ (g Æ $)
[ v2
f [ 3,
A La2 3
1
A La2 2
o
Lt2
Lt2
A
A
(45)
where P3 is determined by requiring that $ Æ g \ 0.
3
For simplicity,
we evaluate the deformation3su†ered by a
wave packet traveling in the positive a -direction. It proves
convenient to transform to coordinatesA
a\a [v t , q\t .
(46)
A
A
As given in equation (40), the unperturbed wave packets in
these coordinates have the forms
g` \ g`(a , a) , g~ \ g~(a , a ] 2v q) .
(47)
1
1 M
1
1 M
A
The equation of motion for g in the new variables, q and
3
A 4 (a , a), reads
M
L
L L
L
L L
[ 2v
g \ [(g Æ $)
[ 2v
A La 3
1
A La
Lq Lq
Lq Lq
A
B
A
]$
P
B
d3A@ $g~ : $g` $P3
1
1 [ 3,
4n o A [ A@ o
o
(48)
where the primes indicate dummy integration variables and
the choice of superscripts applied to g on the right-hand
1
side of the equation is dictated by the requirement
that the
di†erential operators acting on the integral do not kill it ;
one g` and one g~ must appear in the integral over d3A@
1
1
14 SG missed these distortions because they studied wave packet interactions in which the wave packets were localized in g ; for these *gB \ 0,
1
and third-order couplings vanish even in Eulerian coordinates. This
is
consistent with the point made in footnote 5 of SG (p. 616), that resonant
coupling coefficients are independent of the variables used. Of course, for
this to be true, one must be in a regime in which perturbation theory is
valid, which obtains only for wave packets localized in g.
15 NBÏs discovery was made using reduced MHD.
686
GOLDREICH & SRIDHAR
because the derivatives L/Lq and (L/Lq [ 2v L/La) annihilate
A
g` and g~, respectively.16
1 Further
1 expansion of the right-hand side of equation (48),
obtained by writing g \ g` ] g~, results in two terms.
1
1
1
Rather than carry the cumbersome pressure term through
the remainder of our calculation, we discard it and replace
g on the left-hand side of the equation of motion by g8 , to
3
3
remind us that its longitudinal part must be subtracted o†
at a later stage :
A
B
A
B
L
L
L
L L
[ 2v
g8 \ [
[ 2v
(g~ Æ $)
A La 3
A La 1
Lq
Lq Lq
L
$
Lq
]
A
]
P
BP
d3A@ $g~ : $g`
1
1 .
4n o A [ A@ o
(49)
Although we have used coordinates, (a,q), moving with the
positive wave packet, we emphasize that the equation of
motion (eq. [49]) remains symmetric in g8 ` and g8 ~ , i.e., the
3
3
action of the Ðrst term on a (^) wave packet
is identical
to
that of the second term on a (<) wave packet. To determine
the perturbations su†ered by a (]) wave packet, we set
g8 \ g8 ` on the left-hand side.
3 If the
3 Ðrst term were the only perturbing interaction, we
could peel o† the operator (L/Lq [ 2v L/La) from both
sides. Integration over time would yield A
P
P
`=
d3A@ $(Lg~/Lq) : $g`
1
1 , (50)
dq(g~ Æ $)$
1
4n
o A [ A@ o
~=
which describes the four-wave interactions (cf. eq. [23]) that
form the basis of the weak turbulence theory of SG.
The third-order Eulerian velocity perturbation consists of
both kinematic and dynamic terms. An explicit expression
follows from equation (19) :
*g8 ` \ [
3a
Lg
Lg
1
1 ] (g Æ $ )(g Æ $ ) 1
¿ (x, t) \ g g : $ $
1 x 1 x Lt
3
2 1 1 x x Lt
Lf
Lg
Lf
Lg
[ (g Æ $ ) 2 [ (f Æ $ ) 1 ] 3 ] 3 .
1 x Lt
2 x Lt
Lt
Lt
(51)
The Ðnal term is the sole dynamic entry. Each of the other
terms is constructed from g without use of the equation of
1
motion. In particular, f is obtained
from equation (32).
3
Let us estimate the fractional distortion su†ered by a
positive wave packet after traveling a distance z ? l. Among
the plethora of kinematic terms, consider only
Lg`
1
1 .
*¿` \ *(g~g~) : $ $
1 1
x x Lt
3
2
(52)
To order of magnitude, both this term and the dynamic
term contribute
A B
lv~ 2 z
d4v`
j D j
,
jv
l
v`
A
j
the same order of magnitude as the fractional Eulerian distortion given by equation (17).17
To make contact with the weak four-wave cascade, we
suppose that the velocity spectrum of the negatively
directed waves is cut o† at k D L~1 > l~1.18 Then the disz
tortions build up coherently over distance L . Over longer
distances, z ? L , they add with random phases, implying
A B AB
lv 2 L z 1@2
d4v
j
jD
.
(54)
jv
l L
v
A
j
Cascade occurs when d4v /v D 1 ; thus
j j
jv 4 l
A
.
(55)
N D
j
lv
L
j
But for the extra factor of l/L , this is identical to equation
(26), whereupon, provided that N ? 1, the spectrum of the
j
weak four-wave cascade, given in equation
(25), follows.
Next we return to investigate the second term in equation
(49). Imagine that the Ðrst term was switched o†. Then
remove a L/Lq from both sides, leaving the left-hand side in
the form (L/Lq [ 2v L/La)g8 ` . We expect the action of L/Lq
3b
to be subdominant ;Atherefore
A B
d3A@ $g~ : $g`
L
1
1 [ (g` Æ $)
4n o A [ A@ o
Lq 1
L
L
[ 2v
$
A La
Lq
Vol. 485
(53)
16 Proof of the second of these relations requires an integration by parts
to transfer L/La acting on 1/ o A [ A@ o to L/La@ acting on $g : $g .
1
1
Lg8 `
3b ^ [(g` Æ $)$
1
La
P
d3A@ $g~ : $(Lg`/La@)
1
1
,
4n
o A [ A@ o
(56)
implying that the net change in the (]) wave packet is
A B
*
Lg8 `
3b ^ [(g` Æ $)$
1
La
P
d2a@
M $g~ o= :
1 ~=
4n
P
da@
$(Lg`/La)
1
.
o A [ A@ o
(57)
We have written equation (57) in a form that makes
explicit the dependence of the perturbation on only the net
displacement induced by the ([) wave packet. In other
words, the perturbation induced in a wave packet is proportional to the amplitudes of the k \ 0 Fourier comz
ponents of oppositely directed wave packets
: these terms
were Ðrst discovered by NB in the limit of reduced MHD.
We may describe the interactions by the following kind of
resonant four-wave process :
k \k ]k ]k , u \u ]u ]u ,
(58)
1
2
3
4
1
2
3
4
wherein one wave may be induced (by an oppositely
directed wave) to split into two, or two waves could equally
well be induced to combine into one. Manipulating the
resonance relations leads to k , k , k
positive (say),
2,z Thus
3,z u \ u ] u
while k approaches zero from1,z
below.
2 and3
allows 4,z
for changes in the frequencies of wave 1packets,
harmonics, as well as subharmonics, are generated by this
process. These four-wave interactions require the energy
spectrum of the negatively directed packets to be Ñat near
k \ 0 ; thus they do not arise in weak turbulence. Are they
ofz importance to intermediate turbulence ? To this end, let
us make an order-of-magnitude estimate of the pertur-
17 We draw particular attention to the linear dependence on z/l.
18 For simplicity we consider the symmetric situation and drop the ^
signs in the superscripts.
No. 2, 1997
MHD TURBULENCE REVISITED
bation. From equation (57), the perturbation su†ered by the
(]) wave packet upon traveling a distance z D v t is
A
dg` D k2 g` g` *g~ ,
(59)
j
M j j
j
where *g~ 4 g~(j , z) [ g~(j , 0). Using g D l(v /v ),
j
M
M
j
j A
v~
(60)
*g~ D j (lz)1@2 .
j
v
A
Thus the fractional distortion of the positively directed
wave packet is
AB
dv` dg` lv` lv~ z 1@2
j D j D j j
,
(61)
g`
jv jv l
v`
j
A A
j
which is smaller than the corresponding quantity in intermediate turbulence (set n \ 4 in eq. [17], and compare with
the above eq. [61]) by a factor (z/l)1@2 ; thus, harmonic generation is unimportant for intermediate turbulence.19
4.
EVOLUTION OF A WEAK PERTURBATION
Now we address the issue of the physical relevance of the
cascade proposed by SG for weak turbulence. We go on to
show that weak turbulence, intermediate turbulence, and
strong turbulence can be consecutive stages of a single turbulent cascade.
Imagine exciting shear Alfven waves (isotropically on
scale l/L > 1 with amplitude v /v > 1) in a cubical box
l A Ñuid that is threaded
Ðlled with an electrically conducting
by an unperturbed magnetic Ðeld aligned parallel to the
z-axis. Let the box have side length L and assume that its
walls are made of an excellent electrical conductor. Then
Ðeld-line displacements associated with turbulent motions
must vanish at the walls. This boundary condition provides
the cuto† of the energy spectrum for k L \ 1.
z
Suppose that
AB
v 2
l
l > .
(62)
v
L
A
Then Ðeld lines initially separated by scale l maintain this
approximate spacing. This ensures that there are no resonant three-wave interactions in the upper part of the
cascade and that resonant four-wave interactions dominate.
From equation (25), we have
AB
v j 2@3
v
jD l
.
(63)
v
l
v
A
A
The weak cascade grades into the intermediate cascade
when
AB
v 2 j2
j D ,
v
lL
A
which occurs for
A BA B
v 3 L 3@2
j
1D l
.
v
l
l
A
19 However, it might play an important role in strong turbulence.
(64)
(65)
687
For the intermediate cascade,
A B A B AB
v 3@2 L 1@4 j 1@2
v
jD l
.
(66)
v
l
l
v
A
A
The intermediate cascade steepens into the strong cascade
when
j
v
jD ,
l
v
A
which takes place at
A BA B
v 3 L 1@2
j
2D l
.
v
l
l
A
Within the inertial range of the strong cascade,
(67)
(68)
A BA B AB
v 2 L 1@3 j 1@3
v
jD l
.
(69)
v
l
l
v
A
A
The complete three-dimensional inertial-range spectrum
is given by
E(k , k ) D l3v2
z M
2
(k l)~10@3
M
v
L 1@2
l
]
(k l)~3
M
v
l
A
v 2 L 2@3
l
(k l)~8@3
M
v
l
A
g
A BA B
A BA B
G
G
G
l~1 \ k \ j~1 ,
M
1
L~1 \ k \ l~1 ,
z
j~1 \ k \ j~1 ,
M
2
for 1
L~1 \ k \ l1 ,
z
k [ j~1 ,
2
for M
L~1 \ k \ (k j )2@3l~1 ,
z
M 2
(70)
for
There are a couple of points worth noting in connection
with the above combined cascade. The intermediate cascade
is conÐned to j [ j [ j , where the ratio
2
1
l
j
(71)
R4 2D
L
j
1
is independent of v /v . As R ] 1 from below, the inertial
A
range of this cascadel shrinks
to zero, exposing a direct transition between the weak and strong cascades. This is the
transition discussed in SG and Goldreich & Sridhar (1995).
5.
DISCUSSION
In a recent paper, Ng & Bhattacharjee (1996) claimed
that (1) in weak MHD turbulence, three-wave interactions
between oppositely directed wave packets are nonzero if the
k \ 0 components are nonzero and (2) three-wave interz
actions
dominate over four-wave interactions. We hope to
have persuaded the reader that (1) is correct so long as
““ weak ÏÏ is altered to ““ intermediate ÏÏ and that (2) is true
only for individual collisions between small-amplitude wave
packets. However, NB deserve full credit for demonstrating
the importance of three-wave interactions, as well as discovering the frequency-changing terms in four-wave interactions. Both are a consequence of a nonzero net
displacement of Ðeld lines, due to the perturbations of wave
packets that have localized velocity and magnetic Ðeld perturbations.
Intermediate turbulence shares with weak turbulence the
property that wave packets are long-lived : interaction times
688
GOLDREICH & SRIDHAR
are much longer than the wave periods. The distinguishing
feature is that perturbation theory is not applicable to intermediate turbulence ; this should be clear from our demonstration that interactions of all orders have the same
strength. Of course, during individual collisions, three-wave
interactions dominate over all higher order interactions.
However, as described in ° 2.2.2, for wave packets localized
in velocity and magnetic Ðelds, but not in the net displacement of Ðeld lines, subsequent collisions are correlated ; this
makes all n-wave interactions contribute equally to intermediate turbulence. If Iroshnikov and Kraichnan had performed their heuristic estimates taking account of the fact
that there is no parallel cascade for long-lived wave packets,
they would have found the spectrum of the intermediate
cascade. This is one case in which the assumption of isotropy (usually innocuous) is misleading ; the anisotropic
intermediate cascade strengthens on small spatial scales,
whereas the isotropic IK cascade weakens.
We have devoted attention almost exclusively to shear
Alfven (““ sA ÏÏ for brevity) waves, ignoring the dynamics of
pseudo-Alfven (““ pA ÏÏ) waves.20 A generic excitation may be
expected to put power equally into both kinds of waves.
Should not we then study the interactions of the pA waves
among themselves, as well as with sA waves ? Will this
modify the cascades we derived for the sA waves ? It turns
out that the pA waves are slaved to the sA waves. The
reason is as follows : Suppose that we were following the
distortions su†ered by a positively directed wave packet due
to other, negatively directed ones. The nonlinear interactions are given, to order of magnitude, by some power of
(¿~ Æ $)¿` D (k Æ ¿~)¿`. Because the cascades in MHD
turbulence are anisotropic, we expect that k ? k in the
z wave
inertial range. We note that the polarization ofM an sA
20 The only exception is the proof that there are no three-wave couplings for either type of Alfven wave, in the Lagrangian perturbation theory
of SG.
Frisch, U. 1995, Turbulence (Cambridge : Cambridge Univ. Press)
Goldreich, P., & Sridhar, S. 1995, ApJ, 438, 763
Iroshnikov, P. S. 1963, AZh, 40, 742
Kraichnan, R. H. 1965, Phys. Fluids, 8, 1385
Montgomery, D., & Matthaeus, W. H. 1995, ApJ, 447, 706
Newcomb, W. A. 1962, Nucl. Fusion Suppl., 2, 451
Ng, C. S., & Bhattacharjee, A. 1996, ApJ, 465, 845 (NB)
is essentially along zü Â k , and that of a pA wave is along zü ,
M
and hence estimate that the ““ operator ÏÏ k Æ ¿ D (k v A)
M s
] (k v A) D (k v A).21 Thus the pA waves are slaved to
z p
M s
the sA waves ; in a later paper, we will derive kinetic equations and demonstrate that the spectrum of the pA waves is
identical to that of the sA waves for weak, intermediate, and
strong turbulence.
How might an energy spectrum with a cuto† below
k L D 1 arise ? Two related possibilities come to mind. The
z
Ðrst involves a thought experiment that could be realized
computationally ; we described this in ° 4. A more natural
setting might be the atmosphere of a massive star that possesses a strong external dipole Ðeld.22 The energy spectrum
of waves in the stellar magnetosphere would be cut o† on
scales longer than the length of the Ñux tubes that link the
northern and southern magnetic hemispheres. The cuto† at
small k necessary for weak turbulence makes it, in general,
z
less applicable than intermediate turbulence. However, the
limited inertial ranges of both weak and intermediate cascades suggests that neither is likely to Ðnd much application
in nature. The critically balanced cascade, proposed by
Goldreich & Sridhar (1995) for strong MHD turbulence,
remains the most likely candidate for interstellar turbulence.
Financial support for this research was provided by NSF
grant 94-14232. P. G. beneÐted from visits to the Institute
for Advanced Study and the Canadian Institute for Theoretical Astrophysics. We thank Omer Blaes for comments
that improved the clarity of the manuscript.
21 If the amplitude of the pA wave is very much larger than the sA wave,
we could imagine that (k v A) \ (k v A), but this is an unlikely situation.
M s rules zout
p this possibility in cases in which
However, a detailed analysis
both waves have comparable amplitude on the outer scale.
22 Stars with masses in excess of a few solar masses have radiative
atmospheres with little mass motion.
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Press)
Shebalin, J. V., Matthaeus, W. H., & Montgomery, D. 1983, J. Plasma
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Sridhar, S., & Goldreich, P. 1994, ApJ, 432, 612 (SG)
Zakharov, V. E., LÏvov, V. S., & Falkovich, G. 1992, Kolmogorov Spectra
of Turbulence, Vol. 1, Wave Turbulence (Berlin : Springer)